A038107 Number of primes < n^2.

0, 0, 2, 4, 6, 9, 11, 15, 18, 22, 25, 30, 34, 39, 44, 48, 54, 61, 66, 72, 78, 85, 92, 99, 105, 114, 122, 129, 137, 146, 154, 162, 172, 181, 191, 200, 210, 219, 228, 240, 251, 263, 274, 283, 295, 306, 319, 329, 342, 357, 367, 378, 393, 409, 421, 434, 445, 457, 474

Offset: 0

Author

Joe K. Crump (joecr(AT)carolina.rr.com)

Keywords

• nonn

Comments

Also number of primes <= n^2 since n^2 is not prime.

Also the number of primes contained within an n X n square spiral. - William A. Tedeschi, Mar 03 2008

For large n, these numbers closely approximate the sum of primes less than n. For example, n = 10^10, sum of primes < n = 2220822432581729238. The number of primes < (10^10)^2 = 10^20 = 2220819602560918840. The error is 0.0000012743... The derivation of this is in the link Sum of Primes. - Cino Hilliard, Jun 09 2008

a(n) - A000720(n) = A073882(n) - A010051(n) = A117490(n). - Reinhard Zumkeller, May 20 2010

A061265(a(n)) = 1 for n > 1. - Reinhard Zumkeller, Apr 15 2013

From Zhi-Wei Sun, Feb 17 2014: (Start)

Conjecture:

(i) The sequence a(n)^(1/n) (n = 3, 4, ...) is strictly decreasing (to the limit 1).

(ii) If n > 0 is not among 25, 35, 44, 46, 105, then the interval [a(n), a(n+1)] contains at least one prime. (End)

A classical conjecture of Legendre asserts that a(n) < a(n+1) for all n > 0.

Conjecture: All the numbers Sum_{i=j,...,k} 1/a(i) with 1 < j <= k have pairwise distinct fractional parts. - Zhi-Wei Sun, Sep 24 2015

Examples

			a(2)=2 because the only primes < 4 are 2 and 3.
		

References

• Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187. (See Conjectures 2.14-2.16.)

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Cino Hilliard, Sum of Primes.
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
• Wikipedia, Legendre's conjecture.

Crossrefs

Cf. A014085 (first differences), A111208, A194189, A262408, A262443, A262447, A262462.

Programs

Haskell

a038107 0 = 0
a038107 n = a000720 $ a000290 n
-- Reinhard Zumkeller, Apr 15 2013, Nov 01 2011

Maple

A038107 := proc(n) numtheory[pi]( n^2) ; end: seq(A038107(n),n=0..100) ; # R. J. Mathar, Jun 22 2009

Mathematica

Table[PrimePi[n^2], {n, 0, 100}] (* Ray Chandler, Oct 22 2005 *)

PARI

a(n)=primepi(n^2) \\ Charles R Greathouse IV, Apr 26 2012

Sage

[prime_pi(n^2) for n in range(0, 59)] # Zerinvary Lajos, Jun 06 2009

Formula

a(n) = A000720(A000290(n)).

a(n) ~ 1/2 * n^2/log n. - Charles R Greathouse IV, Apr 26 2012

Extensions

Extended by Ray Chandler, Oct 22 2005